Zipf's rank-size rule, lognormal distribution, and Gibrat's urban growth models are considered as summarizing fundamental properties of systems of cities. In this article, they are used as statistical benchmarks for comparing the shapes of urban hierarchies and evolutionary trends of seven systems of cities in the world including BRICS, Europe, and United States. In order to provide conclusions that avoid the pitfalls of too small samples or uncontrolled urban definitions, these models are tested on some 20,000 urban units whose geographically significant delineations were harmonized in each country over 50 years between 1960 and 2010. As a result, if the models appear not always statistically valid, their usefulness is confirmed since the observed deviations from empirical data remain limited and can often be interpreted from the geohistorical context of urbanism proper to each world region. Moreover, the article provides new free software which authorizes the reproducibility of our experiments with our data bases as well as with complementary data.